Levi L. Conant Lecture Series

Alex Kontorovich
2014 AMS Levi L. Conant Prize Recipient

On spheres and bowls of integers

Throughout the last 2000 or so years, circle and sphere packings have delighted curious minds from all over the globe, inspired by their fractal beauty and infinite symmetry. We will revisit this fascinating story, meeting along the way such luminaries as the Great Greek Geometer Apollonius, the French philosopher Rene Descartes, the German co-inventor of calculus Gottfried Willheim Leibniz, and even Frederick Soddy, the English radiochemist and Nobel laureate.

Alex Kontorovich received his BA from Princeton in 2002, and his PhD from Columbia in 2007. Before moving to Rutgers in 2014, he taught at Brown, Stony Brook, and Yale, and held visiting positions at Harvard, ETH (Zurich), and IAS (Princeton). His research concerns problems at the intersection of number theory, geometry, and chaotic dynamics. In addition to the AMS Conant Prize, he is the recipient of an Alfred P. Sloan Research Fellowship, and an NSF CAREER award.

John Huerta
2013 AMS Levi L. Conant Prize Recipient

Symmetry in Mathematics and Physics

Deep at the heart of any discipline lies the idea of “symmetry.” We will explore the fascinating tale of symmetry, from its codification into a powerful tool called group theory by mathematicians in the 19th century, to its rise to the center of fundamental physics in the 20th century, and its evolution and influence today. Group theory begins with intuitive, pictorial ideas of what it means to have symmetry. In the 1830s, the 20 year old genius Evariste Galois invented group theory and turned it into a powerful tool in pure mathematics, but one devoid of apparent practical use. Much later, after decades of mathematical development, Albert Einstein introduced symmetry to physics with his theory of relativity. Yet it was only in the latter half of the 20th century that we discovered the true importance of symmetry in physics: particle physicists discovered it at the heart of the laws of nature, essentially giving our most basic laws their form. It has continued to have a central place ever since, and today, new mathematical ideas about symmetry, with exotic names like “quantum groups” and “higher categories,” may be poised to revolutionize the physics of the 21st century.

John Huerta earned his PhD in 2011 in mathematics from the University of California, Riverside. Huerta is a mathematical physicist, currently a postdoctoral fellow at Centre for Mathematical Analysis, Geometry, and Dynamical Systems in Lisbon. In 2013 he shared the Levi L. Conant Prize with his advisor, John Baez, at University of California, Riverside, for the paper “The Algebra of Grand Unified Theories.” Bulletin of the American Mathematical Society 47:483-552, 2010.

David Vogan
2011 AMS Levi L. Conant Prize Recipient

The Character Table of E8(ℝ)

In 2007, a group of about 20 mathematicians completed the computation of character tables for all the real forms of the exceptional Lie groups, using algorithms introduced by Kazhdan and Lusztig nearly 30 years ago. In the case of the 248-dimensional group called E8(ℝ), the character table (in a very compressed form) occupies about 50 gigabytes of disk space. I’ll talk about several (closely related) questions:

Since these groups have infinitely many conjugacy classes and infinitely many representations, how can one write a character table in finite terms?

What assurance is there that these enormous tables are correct?

How can one extract from them information that a human can understand and find interesting?

As a corollary of these investigations, I will also try to shed some light on the question of whether computers are animated by a demonic malevolence toward humanity.

David Vogan has been a member of the MIT faculty since 1979. He received his PhD from MIT in 1976, under the direction of Bertram Kostant. Most of his work concerns representation theory of Lie groups. He has written papers and books with 13 separate co-authors (an approach he recommends for its effect on Erdös number, for relief of writer’s cramp, for looking smarter, and for enjoying mathematics). He is a member of the American Academy of Arts and Sciences.

Bryna Kra
2010 AMS Levi L. Conant Prize Recipient

Patterns in the primes

In 2004, Ben Green and Terence Tao made a stunning breakthrough, showing that the primes contain arbitrarily long arithmetic progressions.

Perhaps even more impressive is the fusion of methods and results from number theory, ergodic theory, harmonic analysis, and combinatorics used in its proof. The starting point for their proof is the celebrated theorem of Endre Szemerédi from the 1970's: a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Shortly thereafter, Hillel Furstenberg gave a new proof of this theorem, uncovering beautiful connections between dynamics and additive combinatorics. More recently, Timothy Gowers gave a new proof of Szemerédi's Theorem vastly improving quantitative bounds in the finite version. Although the various proofs, Szemerédi's, Furstenberg's, and Gowers's, seem to use very different methods, they have several features in common: in each, a key idea is the dichotomy in the underlying space between randomness and structure. Green and Tao's proof draws on all of these proofs and exploits such a dichotomy. The talk will be an overview of the connections between these topics, with a focus on recent developments.

Bryna Kra earned her undergraduate degree from Harvard University in 1988 and her PhD from Stanford in 1995. Before her appointment to Northwestern University in 2004, she held postdoctoral positions at the Hebrew University of Jerusalem, the University of Michigan, the Institut des Hautes Études Scientifiques, and Ohio State University, and was an assistant professor at Pennsylvania State University. Kra works in dynamical systems and ergodic theory, with a focus on problems related to combinatorics and number theory. She was an invited speaker at the 2006 International Congress of Mathematicians, was awarded a Centennial Fellowship, also in 2006, and was awarded the Conant Prize in 2010. Kra organizes a mentoring program for women in mathematics at Northwestern, runs a math enrichment program for children at a local elementary school, and is currently chair of the Northwestern University math department.

John W. Morgan
2009 AMS Levi L. Conant Prize Recipient

In 1904 Poincaré published a long exposé on 3-dimensional manifolds. At the end of the exposé he asked if every closed (i.e., compact without boundary) 3-dimensional manifold that is simply connected, meaning that every loop in the space can be continuously contracted within the space to a point, was topologically equivalent to the 3-sphere. The conjecture that this was indeed the case became known as the Poincaré Conjecture. It formed the central motivating problem in the next one hundred years of development in topology. It was generalized to higher dimensions, where it was solved (affirmatively) in 1960 by Smale. It was generalized to a question about all 3-manifolds, not just simply connected ones, and many of these questions were resolved in the 1960s by Waldhausen. In 1982 Thurston proposed a much stronger conjecture called the Geometrization Conjecture, which describes all 3-dimensional manifolds in terms of especially nice geometric pieces and includes as a very special case the Poincaré Conjecture.

In this talk I will begin by sketching some of the history of this problem and describing some of the methods of attempted solution. Then I will explain Thurston’s Conjecture and show how it suggests a more differential geometric and analytic approach to the study of the topology of 3-dimensional manifolds. At the end I will introduce Hamilton's Ricci flow equation and indicate how this parabolic, heat-type equation can be used to establish the Geometrization Conjecture and hence the Poincaré Conjecture.

John Morgan was named director of the Simons Center for Geometry and Physics at Stony Brook University in September 2009. He served as professor at Columbia University (1974–2009), assistant professor at MIT (1972–1974), and instructor at Princeton University (1969–1972). He received his PhD from Rice University in 1969.

He has held visiting positions at the Institut des Hautes Études Scientifiques in Paris, Université de Paris-Sud, MSRI in Berkeley, Harvard University, the Institute for Advanced Study in Princeton, and Stanford University. His area of expertise is topology, especially its connections with geometry, algebraic geometry and mathematical physics.

Morgan is the author of more than 50 journal articles and several books; his latest is Ricci Flow and the Poincaré Conjecture. He is a member of the National Academy of Sciences.

Expander graphs are extremely useful objects. In computer science, their applications range from network design, computational, derandomization error correction, data organization, and more. In mathematics they are used in topology, group theory, game theory, information theory, and naturally, graph theory. I plan to explain what expanders and their basic properties are, and survey the quest to explicitly construct them. I’ll focus on the recent combinatorial constructions, via the “zig-zag” product, and how these can go beyond the bounds achieved by algebraic methods. I'll also demonstrate some of the applications.

This talk is accessible to graduate students with no special background in Math and Computer Science.

Avi Wigderson received his BSc in computer science from Technion in 1980, and his PhD from Princeton in 1983. He served on the faculty at the Hebrew University in Jerusalem from 1986 to 2003, and is currently a member of the mathematics faculty at the Institute for Advanced Study at Princeton. His research interests lie principally in complexity theory, algorithms, randomness, and cryptography. His honors include the Nevanlinna Prize for outstanding contributions in mathematical aspects of information sciences (1994), the ICM Plenary Lecture in Madrid, Spain (2006), the AMS Conant Prize in 2008, and the Gödel Prize in 2009.

Brian Conrey
2008 AMS Levi L. Conant Prize Recipient

The Riemann Hypothesis - A million dollar mystery

The famous Riemann Hypothesis is nearly 150 years old. It was on Hilbert's list of 23 problems in 1900 and now it is on the Clay list of Millennium Prize Problems, and has a one million dollar reward for its solution.

Many people regard it as the most important unsolved problem in all of mathematics. In this talk we will explain exactly what the Riemann Hypothesis is and give some of the colorful history that has grown up around efforts to solve it.

Brian Conrey is the founding Executive Director of the American Institute of Mathematics. He received his PhD from the University of Michigan, has served on the faculties of University of Illinois and Oklahoma State University, and was a member of the Institute for Advanced Study. He serves as an editor of the Journal of Number Theory and is also active in several outreach programs for high school students interested in mathematics.

Jeffrey Weeks
2007 AMS Levi L. Conant Prize Recipient

The Shape of Space

When we look out on a clear night, the universe seems infinite. Yet this infinity might be an illusion. During the first half of the presentation, computer games will introduce the concept of a “multiconnected universe.”

Interactive 3D graphics will then take the viewer on a tour of several possible shapes for space. Finally, we'll see how recent satellite data provide tantalizing clues to the true shape of our universe.

The only prerequisites for this talk are curiosity and imagination. Middle school and high school students, people interested in astronomy, and all members of the WPI community are welcome to attend.

Jeffrey Weeks, an independent scholar residing in New York state, has received the 2007 AMS Levi L. Conant Prize for his article "The Poincaré Dodecahedral Space and the Mystery of the Missing Fluctuations," Notices of the AMS, June/July 2004.

In this article, together with an earlier one, "Measuring the Space of the Universe" (Notices, December 1998), co-authored with Neil Cornish, Weeks explains how extremely sensitive measurements of microwave radiation across the sky provide information about the origins and shape of the universe. Weeks discusses what kind of shape our universe could have. The three possibilities are a spherical universe, a Euclidean universe, or a universe that is a hyperbolic 3-manifold.

"Weeks has explained the mathematics behind models whose validity cosmologists debate while waiting for more experimental evidence... Weeks has given a rare glimpse into the role of mathematics in the development and testing of physical theories," the prize citation says.

In 1999 Weeks was awarded a MacArthur "genius" fellowship and now works as a freelance mathematician. He is well known for his geometry and topology software, as well as for his work in cosmology.

Levi Leonard Conant, 1857-1916, Longtime Faculty Member at WPI

Levi L. Conant was a mathematician and educator who spent most of his career as a faculty member at Worcester Polytechnic Institute. He was head of the mathematics department from 1908 until his death, and served as interim president from 1911 to 1913.

Conant was noted as an outstanding teacher, and an active scholar. He published a number of articles in scientific journals and wrote four textbooks: The Number Concept: Its Origins and Development (1896), Original Exercises in Plane and Solid Geometry (1905), Five-Place Logarithmic and Trigonometric Tables (1909), and Plane and Spherical Trigonometry (1909).

Upon his premature death, in 1916, a large bequest was made to The American Mathematical Society, which established the Levi L. Conant Prize, awarded annually to recognize the best expository paper published in either Notices of the AMS or the Bulletin of the AMS during the previous five years.